A highly effective catalytic conversion of CO2 and CO into methanol has been investigated by the multi-functional catalysts composed of Cu-Zn oxides as the main components with the modification of a low concentration of precious metals and gallium oxide. The desired reduced state of the catalyst metal oxides for exhibiting the optimum catalytic performance could be controlled by both the hydrogen spillover through the precious metal parts and the inverse-spillover from the Ga parts. As a result, an extraordinary high space-time yield of methanol, 1300 g/1.h, could be realized under conditions of 270 degrees C, 80 atm, SV 18 800 h(-1) with 22.0% CO2 conversion to methanol. Furthermore, the catalyst also exhibited a very high performance in CO hydrogenation, and gave a space-time yield of methanol as high as 6340 g/1.h under conditions of 270 degrees C, 80 atm, SV 37 600 h(-1) with 39.4% CO conversion to methanol.